By the Weierstrass theorem, we know that on an interval $[a,b]$, any continuous function $f(x)$ can be approximated to arbitrary precision by polynomials. In terms of the $L_\infty$ norm, it means that the set of polynomials is dense in the space of continuous functions $C[a,b]$.
The question is, for given continuous function $f$, suppose we want a polynomial $P(x)$ of degree no higher than $N$ so as to minimize the distance between $f$ and $P$,
$$ dist(f, P ) = max_{a\leq x\leq b} |f(x) -P(x) | . $$
How to find this polynomial? The difficulty is that we have not a Hilbert space.